topological quantum computation - department of mathematical
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Topological quantum computationTopological quantum computationJiri ValaJiri Vala
Department of Mathematical PhysicsDepartment of Mathematical PhysicsNational University of Ireland at National University of Ireland at MaynoothMaynooth
NUI MAYNOOTHNUI MAYNOOTH
Tutorial Presentation, Symposium on Quantum Technologies, UniverTutorial Presentation, Symposium on Quantum Technologies, University of Cambridge, August 29, 2006sity of Cambridge, August 29, 2006
IntroductionIntroductionTopological quantum computation and algorithmsTopological quantum computation and algorithms
•• Links and knotsLinks and knots•• Jones polynomialJones polynomial
•• Braid groupBraid group•• Approximating Jones polynomialApproximating Jones polynomial
Topological phases and natural faultTopological phases and natural fault--tolerancetolerance•• Topological quantum field theoryTopological quantum field theory
•• Spectral properties of topological phasesSpectral properties of topological phases
AnyonsAnyons
Physical realizationPhysical realization•• Fractional quantum Hall systemsFractional quantum Hall systems
•• Lattice models of topological phasesLattice models of topological phases
OutlineOutline
Quantum computationQuantum computation
NP-complete• 3-SAT
P
BQP• factorization
NP• graph isomorphism
#P• Jones polynomialtractable
on quantumcomputer
Certain computational complexity classes and their (rather empirical) relations
Non-deterministic Polynomial
Bounded-error Quantum Polynomial
Polynomial
tractable on
classicalcomputer
ChallengesChallengesTheory: Theory: algorithmsalgorithms, computational complexity, , computational complexity, ……Implementations:Implementations: faultfault--tolerance, tolerance, ……
PSPACE
Topological quantum computationTopological quantum computation•• is a unique QC model (though equivalent to standard QCM) => newis a unique QC model (though equivalent to standard QCM) => new algorithmsalgorithms•• natural fault tolerancenatural fault tolerance
NP-complete• 3-SAT
P
BQP• factorization
NP
• graph isomorphism
#P• Jones polynomialapproximation Freedman et al.,
Commun. Math. Phys. 227, 587 and 605 (2002)
Aharonov, Jones, Landau, STOC’06, quant-ph/0511069
tractableon
quantumcomputer
Links and knotsLinks and knotsLinkLink•• a finite family of disjoint, smooth, a finite family of disjoint, smooth, closed curves in closed curves in RR33 or equivalently or equivalently in Sin S33
KnotKnot•• a link with one componenta link with one component
Knot complexity grows very fast:Knot complexity grows very fast:# of crossings # of knots# of crossings # of knots
00 1133 1144 1155 2266 3377 7788 212199 3636
http://www.pims.math.ca/knotplot/zoo/
Jones polynomialJones polynomial•• Laurent polynomial in tLaurent polynomial in t1/21/2,V,VLL(t)(t)•• invariant of links and knots under invariant of links and knots under isotopyisotopy•• #P#P--hard combinatorial problemhard combinatorial problem
-- classical algorithm is exponential in the number of crossingsclassical algorithm is exponential in the number of crossings
Jones, Bull. AMS 12, 103 (1985)
tt--11 V V -- t V t V = (t= (t1/21/2 -- tt--1/21/2) V ) V skein relationskein relation
trivial knottrivial knot
V V = 1 = 1 Examples:Examples:
= = -- (t(t--1/21/2 + t+ t1/21/2) ) V V
= t + t= t + t33 -- tt44V V right hand trefoil knot
“unlink”
Braid group Braid group BBnn Artin, Ann. Math. 48, 101 (1947)
A braid group for n strands (particles) hasA braid group for n strands (particles) hasn generators {1, n generators {1, σσ11, , …… , , σσnn--11} which} whichsatisfy:satisfy: σσiiσσjj = = σσjjσσii for |j for |j -- i| > 1i| > 1 … …
1 i i+1 i+2i-1 n
ExampleExample: : σσii
Closures of braids are links and knotsClosures of braids are links and knotsplat closureplat closure trace closuretrace closure== YangYang--BaxterBaxter
equationequation
trefoil knottrefoil knot
i i+1 i+2 i i+1 i+2
σσiiσσi+1i+1σσii = = σσi+1i+1σσiiσσi+1i+1
Exchanging particles on a plane is not permutation but braiding:Exchanging particles on a plane is not permutation but braiding:
==
A braid group can be usefully represented A braid group can be usefully represented in in TemperleyTemperley--LiebLieb algebra algebra TLTLnn(d(d) which) whichpermits a unitary representation (permits a unitary representation (qubitsqubits!!!)!!!)
time
xy
clockwise counterclockwise
Approximating Jones polynomialApproximating Jones polynomial
There is an efficient, explicit and simple quantum algorithm to There is an efficient, explicit and simple quantum algorithm to approximate Jones approximate Jones polynomial for all t = epolynomial for all t = e22ππi/ki/k::
Aharonov, Jones, Landau, STOC’06, quant-ph/0511069
Theorem 3Theorem 3Approximating the Jones polynomial of the plat closure (Th.2) isApproximating the Jones polynomial of the plat closure (Th.2) is BQPBQP--complete.complete.
Aharonov, Arad, quant-ph/0605181
Theorem 1Theorem 1For a given braid B with n strands and m crossings, and a given For a given braid B with n strands and m crossings, and a given integer k, there is a integer k, there is a quantum algorithm which is polynomial in n, m, k which with all quantum algorithm which is polynomial in n, m, k which with all but exponentially small but exponentially small probability, outputs a complex number r with |r probability, outputs a complex number r with |r –– VVBBtrtr(e(e22ππi/ki/k)| < )| < εεddnn--11 where d = where d = --AA22 –– AA--22, , and and εε is inverse polynomial in n, m, k.is inverse polynomial in n, m, k.
Theorem 2Theorem 2For a given braid B with n strands and m crossings, and a given For a given braid B with n strands and m crossings, and a given integer k, there is a integer k, there is a quantum algorithm which is polynomial in n, m, k which with all quantum algorithm which is polynomial in n, m, k which with all but exponentially small but exponentially small probability, outputs a complex number r with |r probability, outputs a complex number r with |r –– VVBBplpl(e(e22ππi/ki/k)| < )| < εεdd3n/23n/2/N where /N where d = d = --AA22 –– AA--22, and , and εε is inverse polynomial in n, m, k (N is an exponentially large fis inverse polynomial in n, m, k (N is an exponentially large factor).actor).
the trace closure case
the plat closure case
WocjanWocjan, Yard, quant, Yard, quant--ph/0603069ph/0603069
Freedman et al., Commun. Math. Phys. 227, 587 and 605 (2002)
Topological quantum computationTopological quantum computation
ReadoutReadout
ComputationComputation
InitializationInitialization
braidingbraiding time
time
trefoil knottrefoil knot
2D quantum system2D quantum system
excitations: nonexcitations: non--abelianabelian anyonsanyons
fusionfusion
Topological phaseTopological phasevacuumvacuum
•• is a unique QC model (though equivalent to standard QCM) => newis a unique QC model (though equivalent to standard QCM) => new algorithmsalgorithms•• natural fault tolerancenatural fault tolerance
Topological phases: effective theoryTopological phases: effective theory
Freedman,et al., CMP 227, 605 (2002)
•• their effective description is given by topological quantum fietheir effective description is given by topological quantum field theoryld theory(3 dimensional) defined e.g. by the (3 dimensional) defined e.g. by the ChernChern--Simons action:Simons action:
k = 1 k = 1 -- abelianabelian topological phase topological phase -- quantum memoryquantum memoryk k R 2 2 -- nonnon--abelianabeliank = 3, 5 k = 3, 5 …… -- nonnon--abelianabelian and universal and universal -- universal QCuniversal QC
S = kk/4π 4dt d2x εμνρ aμvνaρ
Example: doubled SU(2)Example: doubled SU(2)kk ChernChern--Simons theory (PT invariant theory):Simons theory (PT invariant theory):
gauge field
•• topological phases are phases of twotopological phases are phases of two--dimensional manydimensional many--body quantum systems body quantum systems whose properties depend only on topology of the manifold on whwhose properties depend only on topology of the manifold on whose surface a ose surface a given phase is realizedgiven phase is realized
level of theory (integer) Γ (2+1)D manifold
Witten, Commun. Math. Phys. 121, 351 (1989)
no metric!!!
•• topological phases are invariant with local geometry and topological phases are invariant with local geometry and hence quantum information stored in them is hence quantum information stored in them is invariant with local error processesinvariant with local error processes
no metric, no error!!!
Topological phases: Hamiltonian spectrumTopological phases: Hamiltonian spectrum
•• energy spectrum of matter in a topological phase is characterizenergy spectrum of matter in a topological phase is characterized byed by
finite topologyfinite topology--dependent ground state degeneracy, dependent ground state degeneracy, e.g. for the doubled SU(2)e.g. for the doubled SU(2)kk ChernChern--Simons theory: (k+1)Simons theory: (k+1)22gg
spectral gapspectral gap
Freedman et al. Ann. Phys. 310, 428 (2004)
e.g. in Coulomb gauge, a0 = 0:
•• are ground states of certain strongly correlated manyare ground states of certain strongly correlated many--body quantum systemsbody quantum systems
L = a2v0a1 – a1v0a2
vL vLv(v0a1) v(v0a2)
HH = + - L = 00v0a2v0a1
=>
no metric, no energy!!!
genus
•• excitations of excitations of stray stray anyonsanyons, which may cause , which may cause errors via nonerrors via non--local processes, are exponentially local processes, are exponentially suppressed due to the spectral gap !!!suppressed due to the spectral gap !!!
Topological phases: ground state degeneracyTopological phases: ground state degeneracyExample:Example: abelianabelian anyonsanyons on on torustorus torustorus (genus g=1 surface)(genus g=1 surface)
LL22LL11 ==
LL22
LL11
with opposite sides identified
one one anyonanyon winds clockwise around the other:winds clockwise around the other:
TT22--11TT11
--11TT22TT11 = e= e--i2i2θθ 11
time
TT11
TT11--11
TT22
TT22--11
[T[T11, H] = 0 and [T, H] = 0 and [T2 2 , H] = 0, so T, H] = 0, so T11 ||αα> = > = eeiiαα ||αα>: >: TT11(T(T22)|)|αα> = e> = ei2i2θθTT22TT11||αα> = e> = ei2i2θθeeiiαα(T(T22)|)|αα>>suppose that suppose that θθ is a rational multiple of is a rational multiple of ππ: : θ θ = = π π p/qp/q, then T, then T1 1 has q distinct has q distinct eigenvalueseigenvalues(orbits: (orbits: αα + (2+ (2ππp/q)k(mod 2p/q)k(mod 2ππ), where k=0,1,), where k=0,1,……,q,q--1) and the ground state degeneracy is q.1) and the ground state degeneracy is q.
For genus g surface: For genus g surface: the ground state degeneracy is the ground state degeneracy is qqgg Preskill, Lecture notes
are excitations, quasiparticles, of a topological phase
AnyonsAnyons
OneOne--dimensional dimensional irrepsirreps of of BBnn correspond to correspond to abelianabelian fractional statistics:fractional statistics:
χθ (σ) = eiθ 2 U(1)
Higher dimensional Higher dimensional irrepsirreps correspond to correspond to nonabeliannonabelian fractional statistics: fractional statistics:
Configuration space of n indistinguishable particles in d dimensional space excluding diagonal points D:
- in two spatial dimensions the configuration space is multiply connected
MMnn = (= (RRndnd -- D)/D)/SSnn Leinaas and Myrheim’77Wilczek’82
Exchanging particles on a plane is an element of braid group Exchanging particles on a plane is an element of braid group BBnn::
time
xy
==
χχθθ ((σσ) = ) = eeiiθΛθΛ e.g.e.g. 22 SU(2)SU(2)
• Hilbert space dimension for the trivial sector grows with the number of anyonsas the Fibonacci series: 0, 1, 1, 2, 3, 5, 8 …
Example: Fibonacci Example: Fibonacci anyonsanyons
0 1 1 1
|0>L |1>L
1 0
|NC> = non-computational
J. Preskill, Lecture notes
• are characterized by two possible values of “q-deformed” spin quantum number0 (trivial) and 1
• composition of q-spins is dictated by fusion rules (CFT):1 x 1 = 0 + 10 x 0 = 00 x 1 = 1
• one logical qubit can be constructed with q-spin=1 anyons as follows (reminiscence of encoded universality)
=
are derived from fusion rules and consistency relations between braiding andfusion operations know as pentagon and hexagon equations (quantum groups), the result is:
QC operations with Fibonacci QC operations with Fibonacci anyonsanyons
Bonesteel, et al.’05
-iτ1/2e-iπ/10-τe-iπ/5
-iτ1/2e-iπ/10 -τ-e-i2π/5
e-iπ/5 0
0=
two-qubit operations • braiding between anyons of different logical qubits - requires optimization• analogous to the concept of encoded universality
single-qubit operations:
Topological phases in physical systemsTopological phases in physical systems
•• fractional quantum Hall systemsfractional quantum Hall systemsparticularly promising !!!particularly promising !!!
•• ppxx+ip+ipyy superconductorssuperconductorsSrSr22RuORuO44HeliumHelium--33
•• quantum lattice systems quantum lattice systems atoms in optical latticesatoms in optical latticespolar moleculespolar moleculesJosephsonJosephson--junction arraysjunction arrays
•• rotating Boserotating Bose--Einstein condensatesEinstein condensates
•• nuclear matternuclear matter
DasDas SarmaSarma, et al., Phys. Rev. , et al., Phys. Rev. LettLett. 94, 166802 (2005). 94, 166802 (2005)
DasDas SarmaSarma, et al., Phys. Rev. B 73, 220502 (2006), et al., Phys. Rev. B 73, 220502 (2006)
DuanDuan, et al., Phys. Rev. , et al., Phys. Rev. LettLett. 91, 040902 (2003). 91, 040902 (2003)
MicheliMicheli et al., Nature Phys. 2, 341 (2006)et al., Nature Phys. 2, 341 (2006)
IoffeIoffe et al., Nature 415, 503 (2002)et al., Nature 415, 503 (2002)
SalomaaSalomaa, , VolovikVolovik, Rev. Mod. Phys. 59, 533 (1989), Rev. Mod. Phys. 59, 533 (1989)
Fractional quantum Hall effectFractional quantum Hall effect
VVxx
VVyy
IIxxBB
Longitudinal resistance Longitudinal resistance RRxxxx = = VVxx / I/ Ixx
Transverse (Hall) resistance Transverse (Hall) resistance RRxyxy = = VVyy / I/ Ix x = h / = h / νν ee22
Eisenstein, Eisenstein, StormerStormer, Science 248, 1461 (1990), Science 248, 1461 (1990)
StormerStormer, , TsuiTsui, , GossardGossard, Phys. Rev. , Phys. Rev. LettLett. 48, 1559 (1982). 48, 1559 (1982)Rev. Mod. Phys. 71, S298 (1999)Rev. Mod. Phys. 71, S298 (1999)
TheoryTheorynonabeliannonabelian quantum Hall phases at quantum Hall phases at νν=5/2 and 12/5 =5/2 and 12/5
Experiment Experiment detecting these phases in high mobility samplesdetecting these phases in high mobility samples
XiaXia et al., Phys. Rev. et al., Phys. Rev. LettLett. 93, 176809 (2004). 93, 176809 (2004)
Read, Read, RezayiRezayi, Phys. , Phys. Rev.BRev.B 59, 8084 (1999)59, 8084 (1999)
quantum Hall fluid
Fractional quantum Hall systemsFractional quantum Hall systems•• nonnon--abelianabelian topological phases predicted in FQH systems attopological phases predicted in FQH systems atthe filling the filling νν=5/2 and 12/5 =5/2 and 12/5
•• experimental tests of fractional statistics using Laughlin inteexperimental tests of fractional statistics using Laughlin interferometerrferometer
•• relation between boundary (CFT) and bulk (TQFT) relation between boundary (CFT) and bulk (TQFT) –– holographic principleholographic principle
•• topologically protected topologically protected qubitqubit
edge currents
antidots
gate electrodes DasDas SarmaSarma, Freedman, , Freedman, NayakNayak Phys. Rev. Phys. Rev. LettLett. 94, 166802 (2005). 94, 166802 (2005)
Camino, Zhou, Goldman, Phys. Rev. B 72, 075342 (2005)
t1 t2 t3
Topological phases in quantum lattice systemsTopological phases in quantum lattice systems•• ToricToric code (code (KitaevKitaev))
abelianabelian topological phasetopological phasequantum memoryquantum memory
•• KitaevKitaev honeycomb lattice modelhoneycomb lattice modelabelianabelian topological field in zero magnetic fieldtopological field in zero magnetic fieldnonnon--abelianabelian topological phase in the presence of magnetic fieldtopological phase in the presence of magnetic fieldrealizations proposed using atoms in optical lattices and polar realizations proposed using atoms in optical lattices and polar moleculesmoleculesgraphenegraphene
•• Quantum loop gas model Quantum loop gas model abelianabelian topological phases (k=1)topological phases (k=1)nonnon--abelianabelian topological phases (k=2)topological phases (k=2)concrete physical representation concrete physical representation –– extended Hubbard modelextended Hubbard model
•• Trivalent graph (spinTrivalent graph (spin--1) model 1) model hierarchy of topological phaseshierarchy of topological phases
•• String nets condensation String nets condensation general hierarchy of topological phasesgeneral hierarchy of topological phases
Kitaev, cond-mat/0506438
Fendley, Fradkin, Phys. Rev. B72, 024412
Levin, Wen,Phys. Rev. B71, 045110 (2005)
Freedman et al., Freedman et al., condcond--mat/0309120; mat/0309120; Phys. Rev. Phys. Rev. LettLett. . 94, 066401 (2005).94, 066401 (2005).
Kitaev, quant-ph/9707021Ann. Phys. 303, 2 (2003)
Quantum loop gasQuantum loop gasTwoTwo--dimensional sea of fluctuating loopsdimensional sea of fluctuating loops
formed for example byformed for example by dimersdimers on a quantum lattice with on a quantum lattice with a fixeda fixed background background dimerdimer coveringcovering
- 1/6 filled Kagome lattice- U0 is infinite, excludingdoubly-occupied sites andhence preventing collisions
- potential terms are diagonal in the occupation basis, V and μ terms are color dependent
- tunneling between nearest neighbors, color dependent
- U >> μi, Vij , tij
H = U0 Σi (ni-1)ni +
Σi μini +
U Σi,j ninj +
Σi,jVijninj +
Σi,j tij (ci+cj + cj
+ci )
M. H. Freedman, C. Nayak, K. Shtengel, cond-mat/0309120; PRL 94, 066401 (2005).
Kagome lattice
Dual lattice
Extended Hubbard modelExtended Hubbard model
Topological phase in EHMTopological phase in EHM
quantum loop gasquantum loop gas
M. H. Freedman, C. M. H. Freedman, C. NayakNayak, K. Shtengel, , K. Shtengel, condcond--mat/0309120; PRL 94, 066401 (2005).mat/0309120; PRL 94, 066401 (2005).
3. consistent surgery conditions 3. consistent surgery conditions ––JonesJones--WenzlWenzl idempotentsidempotents, , e.ge.g
where d=2cos(where d=2cos(ππ/(k+2))/(k+2))
Topological conditions:Topological conditions:
ddΨΨ[ ][ ] = = ΨΨ[ ][ ]
1. 1. isotopyisotopy
extensively degenerate ground stateextensively degenerate ground state
2. 2. dd--isotopyisotopy
ddΨΨ[ ][ ] = = ΨΨ[ ][ ]
ΨΨ[ ][ ] = = ΨΨ[ ][ ]
Topological phaseTopological phase
ChallengesChallengesTopological phasesTopological phases
•• microscopic models which are microscopic models which are -- universal for quantum computationuniversal for quantum computation-- based on local interactionsbased on local interactions-- experimentally conceivableexperimentally conceivable
•• physical realizationphysical realization-- quantum Hall systems, etc.quantum Hall systems, etc.
•• classificationclassification
Topological quantum computing operationsTopological quantum computing operations•• braidingbraiding•• measurement (measurement (AharonovAharonov--BohmBohm--likelike experiment)experiment)
AlgorithmsAlgorithms•• approximation of certain statistical mechanical problems (e.g. approximation of certain statistical mechanical problems (e.g. Potts model)Potts model)•• approximation of NPapproximation of NP--complete problemscomplete problems•• graph theoretical problemsgraph theoretical problems
ComplexityComplexity•• BQPBQP--complete problemscomplete problems